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Title: Stochastic Analysis of Advection-diffusion-Reactive Systems with Applications to Reactive Transport in Porous Media

Abstract

We developed new CDF and PDF methods for solving non-linear stochastic hyperbolic equations that does not rely on linearization approximations and allows for rigorous formulation of the boundary conditions.

Authors:
Publication Date:
Research Org.:
University of California, San Diego
Sponsoring Org.:
USDOE
OSTI Identifier:
1149536
Report Number(s):
DOE-UCSD-ER25815
DOE Contract Number:
FG02-07ER25815
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Tartakovsky, Daniel. Stochastic Analysis of Advection-diffusion-Reactive Systems with Applications to Reactive Transport in Porous Media. United States: N. p., 2013. Web. doi:10.2172/1149536.
Tartakovsky, Daniel. Stochastic Analysis of Advection-diffusion-Reactive Systems with Applications to Reactive Transport in Porous Media. United States. doi:10.2172/1149536.
Tartakovsky, Daniel. 2013. "Stochastic Analysis of Advection-diffusion-Reactive Systems with Applications to Reactive Transport in Porous Media". United States. doi:10.2172/1149536. https://www.osti.gov/servlets/purl/1149536.
@article{osti_1149536,
title = {Stochastic Analysis of Advection-diffusion-Reactive Systems with Applications to Reactive Transport in Porous Media},
author = {Tartakovsky, Daniel},
abstractNote = {We developed new CDF and PDF methods for solving non-linear stochastic hyperbolic equations that does not rely on linearization approximations and allows for rigorous formulation of the boundary conditions.},
doi = {10.2172/1149536},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2013,
month = 8
}

Technical Report:

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  • The main objective of this project is to develop new computational tools for uncertainty quantifica- tion (UQ) of systems governed by stochastic partial differential equations (SPDEs) with applications to advection-diffusion-reaction systems. We pursue two complementary approaches: (1) generalized polynomial chaos and its extensions and (2) a new theory on deriving PDF equations for systems subject to color noise. The focus of the current work is on high-dimensional systems involving tens or hundreds of uncertain parameters.
  • A reliability approach for probabilistic modeling of one-dimensional non-reactive and reactive transport in porous media provides two important quantitative results: (1) an estimate of the probability that dimensionless concentration equals or exceeds some specified level and, (2) the sensitivity of the probabilistic outcome to likely changes in each uncertain variable. The reliability approach is particularly attractive because it can incorporate various marginal probability density functions (PDF) for any of the uncertain variables. In this work uncertain variables include: groundwater flow velocity, diffusion coefficient, dispersivity, distribution coefficient, porosity and bulk density. The primary objective is to examine how the probabilistic outcomemore » is influenced by choice of marginal PDF, correlation and magnitude of uncertainty for the variables. Because little information exists concerning the statistical characteristics of these uncertain variables, the investigation assumes a wide range of PDF types and statistical values in order to identify and isolate the most critical issues for further study. Results indicate that, even for very slow mean velocity, the probability estimate for non-reactive transport is most sensitive to uncertain flow velocity. For practical analysis, it appears acceptable to treat dispersivity as a deterministic constant. For non-reactive transport, correlation between flow velocity and diffusion coefficient has a slight impact, but correlation among other combinations of uncertain variables is not important. 34 refs., 10 tabs.« less
  • We investigate Bayesian techniques that can be used to reconstruct field variables from partial observations. In particular, we target fields that exhibit spatial structures with a large spectrum of lengthscales. Contemporary methods typically describe the field on a grid and estimate structures which can be resolved by it. In contrast, we address the reconstruction of grid-resolved structures as well as estimation of statistical summaries of subgrid structures, which are smaller than the grid resolution. We perform this in two different ways (a) via a physical (phenomenological), parameterized subgrid model that summarizes the impact of the unresolved scales at the coarsemore » level and (b) via multiscale finite elements, where specially designed prolongation and restriction operators establish the interscale link between the same problem defined on a coarse and fine mesh. The estimation problem is posed as a Bayesian inverse problem. Dimensionality reduction is performed by projecting the field to be inferred on a suitable orthogonal basis set, viz. the Karhunen-Loeve expansion of a multiGaussian. We first demonstrate our techniques on the reconstruction of a binary medium consisting of a matrix with embedded inclusions, which are too small to be grid-resolved. The reconstruction is performed using an adaptive Markov chain Monte Carlo method. We find that the posterior distributions of the inferred parameters are approximately Gaussian. We exploit this finding to reconstruct a permeability field with long, but narrow embedded fractures (which are too fine to be grid-resolved) using scalable ensemble Kalman filters; this also allows us to address larger grids. Ensemble Kalman filtering is then used to estimate the values of hydraulic conductivity and specific yield in a model of the High Plains Aquifer in Kansas. Strong conditioning of the spatial structure of the parameters and the non-linear aspects of the water table aquifer create difficulty for the ensemble Kalman filter. We conclude with a demonstration of the use of multiscale stochastic finite elements to reconstruct permeability fields. This method, though computationally intensive, is general and can be used for multiscale inference in cases where a subgrid model cannot be constructed.« less
  • One of the most significant challenges facing environmental engineers and scientists is predicting the movement and degradation of chemicals in hierarchical porous media. The distribution of subsurface properties is poorly known because of the inaccessibility of the subsurface environment and the random nature of the geologic deposition process. In addition, the subsurface often possesses distinct physical, chemical and biological hierarchies, which complicates the ability to successfully characterize and thus predict property distributions and processes with information from a limited number of sample locations over a limited number of scales. Knowledge of the spatial structure of microbial populations and activities andmore » the dynamic environmental factors that control this spatial structure are important in characterizing sites for remediation and disposal, and for the ability to effectively deliver nutrients to promote degradation and stabilization. To do so effectively requires a correct theoretical formulation of the problem, implementation of this formulation for predictive purposes, and even more importantly knowledge of what should be measured and how and when to measure it. The contents of this report is as follows: (Section 2) statement of goals, (Section 3) development of nonlocal models for chemical transport with uncertainty in biological, physical and chemical data, (Section 4) a discussion of molecular-scale phenomena of relevance to adsorption and flow in nanoporous materials such as clays, (Section 5) meso and macroscale models of flow in, and deformation of, clays, (Section 6) collaborative efforts with DOE labs, (Section 7) P.I. awards, (Section 8) publications resulting from the research efforts supported through this grant, and finally students supported under this grant.« less
  • This report gives an overview of the work done as part of an Early Career LDRD aimed at modeling flow induced damage of materials involving chemical reactions, deformation of the porous matrix, and complex flow phenomena. The numerical formulation is motivated by a mixture theory or theory of interacting continua type approach to coupling the behavior of the fluid and the porous matrix. Results for the proposed method are presented for several engineering problems of interest including carbon dioxide sequestration, hydraulic fracturing, and energetic materials applications. This work is intended to create a general framework for flow induced damage thatmore » can be further developed in each of the particular areas addressed below. The results show both convincing proof of the methodologies potential and the need for further validation of the models developed.« less